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In the analysis of Markov chains and processes, it is sometimes convenient toreplace an unbounded state space with a "truncated" bounded state space. Whensuch a replacement is made, one often wants to know whether the equilibriumbehavior of the truncated chain or process is close to that of the untruncatedsystem. For example, such questions arise naturally when considering numericalmethods for computing stationary distributions on unbounded state space. Inthis paper, we study general truncation-augmentation schemes, in which thesubstochastic truncated "northwest corner" of the transition matrix or kernelis stochasticized (or augmented) arbitrarily. In the presence of a Lyapunovcondition involving a coercive function, we show that such schemes aregenerally convergent in countable state space, provided that the truncation ischosen as a sublevel set of the Lyapunov function. For stochastically monotoneMarkov chains on $\mathbb Z_+$, we prove that we can always choose thetruncation sets to be of the form $\{0,1,...,n\}$. We then provide sufficientconditions for weakly continuous Markov chains under which generaltruncation-augmentation schemes converge weakly in continuous state space.Finally, we briefly discuss the extension of the theory to continuous timeMarkov jump processes.

On Convergence of General Truncation-Augmentation Schemes for Approximating Stationary Distributions of Markov Chains